Granulometries and Opening Trees

Granulometries constitute one of the most useful and versatile sets of tools of morphological image analysis. They can be applied to a wide range of tasks, such as feature extraction, texture characterization, size estimation, image segmentation, etc., both for binary and for grayscale images. However, for most applications, traditional granulometry algorithms - involving sequences of openings or closings with structuring elements of increasing size - are prohibitively costly on non-specialized hardware. This has prevented granulometries from reaching a high level of popularity in the image analysis community. This paper addresses the computational aspect of granulometries and proposes a comprehensive set of fast algorithms. In binary images, all but the simplest cases (namely linear granulometries based on openings with line segments) require the prior extraction of opening transforms (also referred to as “granulometry functions”). A very efficient algorithm is proposed for the computation of the most useful opening transforms. In grayscale images, linear granulometries are considered first and a particularly efficient algorithm is described. The concept of an opening tree is then proposed as a gray extension of the opening transform. It forms the basis of a novel technique for computing granulometries based on maxima of openings by line segments in different orientations, as well as pseudo-granulometries based on minima of linear openings. Furthermore, opening trees can be used in local granulometry algorithms, thereby making it possible to compute such objects as size transforms directly from grayscale images. Other applications include adaptive openings and closings, as well as granulometric texture segmentation. The efficiency of this set of algorithms greatly increases the range of problems that can be addressed using granulometries. A number of applications are used throughout the paper to illustrate the usefulness of the proposed techniques.